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Flashcards: Describing Exponential vs. Linear Change

The following equation represents the change of a company’s market value since 1993: $M(t)=100,000(0.982)^{t}$ $\displaystyle M(t)=100,000(0.982)^{t}$. Which of the following statements best describes this function?

This company is exponentially shrinking by 1.8% $\displaystyle 1.8%$

This company is exponentially growing by 98.2% $\displaystyle 98.2%$

This company is linearly growing by 98.2% $\displaystyle 98.2%$

This company is linearly shrinking by 1.8% $\displaystyle 1.8%$

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Flashcards: Describing Exponential vs. Linear Change

This company is exponentially shrinking by 1.8% $\displaystyle 1.8%$

This company is exponentially growing by 98.2% $\displaystyle 98.2%$

This company is linearly growing by 98.2% $\displaystyle 98.2%$

This company is linearly shrinking by 1.8% $\displaystyle 1.8%$

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Flashcards: Describing Exponential vs. Linear Change

Your answer:

This company is exponentially growing by 98.2% $\displaystyle 98.2%$

Your answer:

This company is linearly growing by 98.2% $\displaystyle 98.2%$

Your answer:

This company is linearly shrinking by 1.8% $\displaystyle 1.8%$

Correct answer:

This company is exponentially shrinking by 1.8% $\displaystyle 1.8%$

Explanation:

The correct answer is “This company is exponentially shrinking by 1.8% $\displaystyle 1.8%$.” The above function is an exponential function of t because $\displaystyle t$ is an exponent of a base in the function. A linear equation would utilize $\displaystyle t$ as a coefficient, not an exponent. The best way to differentiate between growth and shrinkage in an exponential function is to see if the exponent’s base is greater or less than $\displaystyle 1$. If the base is $\displaystyle 1$, there would be no change in M(t) $\displaystyle M(t)$ regardless of the value of $\displaystyle t$, but if the base is less than $\displaystyle 1$, the company’s value is decreasing. $\displaystyle 1-0.982=0.018=1.8%$ so a loss of 1.8% $\displaystyle 1.8%$ of market value annually.

Flashcards: Describing Exponential vs. Linear Change

Screen shot 2020 09 25 at 11.39.00 am

The above table shows the growth of two plants in cm. Which of the following statements best describes the growth of these two plants?

Plant A shows linear growth while Plant B is exponential.

Both Plant A and B show linear growth

Plant A shows exponential growth while Plant B is linear.

Both Plant A and B show exponential growth

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Flashcards: Describing Exponential vs. Linear Change

Screen shot 2020 09 25 at 11.39.00 am

The above table shows the growth of two plants in cm. Which of the following statements best describes the growth of these two plants?

Your answer:

Plant A shows linear growth while Plant B is exponential.

Your answer:

Both Plant A and B show linear growth

Your answer:

Both Plant A and B show exponential growth

Correct answer:

Plant A shows exponential growth while Plant B is linear.

Explanation:

The correct answer is “Plant A shows exponential growth while Plant B is linear.” Each hour doubles the height of Plant A. Doubling is a characteristic of exponential growth (2t) $\displaystyle (2t)$. Each hour adds $\displaystyle 5$ cm to the height of Plant B. Additive and subtractive trends are characteristic of linear growth (5t) $\displaystyle (5t)$.

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